TITLE: In $C([0,1],\mathbb{R})$, the sup norm and the $L^1$ norm are not equivalent.
QUESTION [6 upvotes]: How does the proof here show that the two norms are not equivalent? We have that in the sup norm $f_n$ converges to $1$, and in the $L^1$ norm $f_n$ converges to $0$, but how does this mean that the two norms are not equivalent?

REPLY [3 votes]: If the two norms were equivalent then the $\|\cdot\|_\infty$-neighborhood $U_{1/2}(0)$  would have to contain a  $\|\cdot\|_1$-neighborhood  $V_\epsilon(0)$ for some $\epsilon>0$. But your example shows that any such $V_\epsilon(0)$ contains functions $f_n$ lying outside of $U_{1/2}(0)$.